Maybe a term you are looking for is "sparse solutions of underdetermined linear systems". At least this would maybe be a good name for your set $V$. It can be viewed as solution set of the following optimization problem:
$$\min||x||_0 \text{ subject to }Ax=b.$$
where $||\cdot||_0$ counts the number of nonzeros So a $P(A,b)$ could be called "convex hull of sparse solutions".
References in this direction are for example Donoho and a survey by Lai.